Some older laptops are known to run hot at the bottom.
Sometimes it is because the battery charger circuitry is trying to charge a weak or troubled battery and also power the laptop at the same time to its power demands.
So questions i have is, is the battery healthy to where it can run for quite a while without being plugged into an outlet? And have you been able to run this laptop directly without battery installed only off of the ac wall adapter powering it to see if it runs cooler at the bottom of it with the battery missing?